A372658 -id:A372658 - cp's OEIS frontend

A372657 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the Fibonacci numbers that appear in the Zeckendorf representation of n are not missing from the dual Zeckendorf representation of a(n).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 16, 21, 18, 20, 22, 23, 24, 25, 28, 29, 31, 26, 27, 32, 30, 33, 34, 35, 36, 38, 40, 37, 42, 39, 46, 48, 47, 51, 53, 41, 43, 44, 45, 56, 49, 50, 52, 54, 55, 57, 58, 59, 62, 63, 65, 60, 61, 66, 64, 67

Offset: 0

Rémy Sigrist, May 09 2024

We consider that a Fibonacci number is missing from the dual Zeckendorf representation of a number if it does not appear in this representation and a larger Fibonacci number appears in it.
The dual Zeckendorf representation is also known as the lazy Fibonacci representation (see A356771 for further details).
This sequence is a permutation of the nonnegative integers with inverse A372658: for any v >= 0, the majority of Fibonacci numbers are not missing from the dual Zeckendorf representation of v, and provide opportunities for v to be chosen, and so v will eventually appear in the sequence.

			The first terms, alongside the Zeckendorf representation of n and the dual Zeckendorf representation of a(n), in binary, are:
  n   a(n)  z(n)    d(a(n))
  --  ----  ------  -------
   0     0       0        0
   1     1       1        1
   2     2      10       10
   3     3     100       11
   4     4     101      101
   5     5    1000      110
   6     6    1001      111
   7     7    1010     1010
   8     8   10000     1011
   9     9   10001     1101
  10    10   10010     1110
  11    11   10100     1111
  12    12   10101    10101

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n
Index entries for sequences that are permutations of the natural numbers

See A372659 for a similar sequence.

Cf. A356771, A361989, A372658 (inverse).

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A372657 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the Fibonacci numbers that appear in the Zeckendorf representation of n are not missing from the dual Zeckendorf representation of a(n).

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